请教martingale与independent increment process

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请问一下martingale与independent increment process之间有什么联系呢？有哪个一定包含哪个的说法没？

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关键词：Independent Martingale Dependent increment Process Process Independent Martingale increment

应该是一个挺基础的问题。。。。。。我概念没掌握好 没弄明白

麻烦大家讲解一下啊 谢谢~

没什么关系啊，

martingale可以是increment independent,也可以不是。举例子的话就是brownian motion is a martingale with increment indenpent,can if we built a process X_t=2*B_t when 0<t<T, and X_t=B_T+B_t when t>=T, this process is gaussian, and you can calculate the correlation and you find that they are not zero.

An independent process is the same, if you take a process like X_t=t, t>0, which is independent increment process, but not a martingale.

Thanks a lot for your post!

One point that I am not quite clear~ :

For the first example, that is X_t=2*B_t when 0<t<T, and X_t=B_T+B_t when t>=T, I understand that it is Gaussian. But I guess it is not a martingale, since the brownian motion is not a U.I. martingale. So E(B_T) would not necessary be E(B_0). (?)

还有关于correlation那部分也不太明白。。。。。。   能否再分析一下？

万谢！

我就是想举一个是martingale但不是independent increment的过程来

我做的那个例子不对，我又算了一遍。

我就是想找一个很简单的例子，找一个martingale,算一下increment的correlation,只要不是零，就不是independent了。又想了一会，一时没有很简单的例子，

不过有个例子是B_t^2-t是个martingale,但不是independent increment.

另，我造的那个例子的确是个martingale，而且E(B_T)=E(B_0=0,这个和U.I martingale没什么关系。

you mean exp(B_t^2-t), the exponential martingale, right?

yes this should be a good example~ thanks!